Question: Given $ m \angle CBD = 6x - 42$, $ m \angle ABC = 6x + 6$, and $ m \angle ABD = 84$, find $m\angle CBD$. $B$ $A$ $D$ $C$
From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {6x + 6} + {6x - 42} = {84}$ Combine like terms: $ 12x - 36 = 84$ Add $36$ to both sides: $ 12x = 120$ Divide both sides by $12$ to find $x$ $ x = 10$ Substitute $10$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 6({10}) - 42$ Simplify: $ {m\angle CBD = 60 - 42}$ So ${m\angle CBD = 18}$.